asset allocation in continuous time

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1、CHAPTER66assetallocationincontinuoustimeInthisChapter...•thewealthprocess•optimizingoverassetholdingsandconsumption•theBellmanequation•theeffectoftransactioncosts66.1INTRODUCTIONInChapter18welookedatwaysinwhichtoinvestoptimallyassumingthatwecannotchangeourm

2、indslater;thisistheone-periodinvestment.Inpractice,wecanchangeourmindsasoftenaswewantaccordingtothewaythatthefutureunveilsitself.Inthischapterweexaminemodelsforcontinuous-timeinvestmentandportfoliorebalancing.MostofthisistakenfromMerton’swork.Oneoftherestri

3、ctionsonrebalancingourportfolioisthecostoftransacting.Attheendofthechapterwetakeabriefpeekattransactioncosts.66.2ONERISK-FREEANDONERISKYASSETTogettheballrolling,let’sfirstlookatinvestinginjusttwoassets,onerisk-freeearningasurerateofrandtheotherriskyfollowing

4、therandomwalkdS=µSdt+σSdX.(66.1)Wearegoingtoinvestafractionw(t)ofourwealthintheriskyassetandtheremainingfraction1−wintherisk-free.Wewillalsochoosetoconsumeorspend,spend,spend,atarateofC(t).Ouraimistodecidehowtosplitourmoneybetweenthetwoassetsandhowmuchtocon

5、sume;thatiswemustdecideonthebestw(t)andC(t).Obviously,wewillhavetooptimizesomeproblemaswe’llshortlysee.1052PartFiveadvancedtopics66.2.1TheWealthProcessHowdoesourwealthevolveastheassetpriceschangeandweconsume?Canwefindastochasticdifferentialequationfortheweal

6、thW?Fromonetimesteptothenextourwealthchangesforthreereasons:TheassetSchangesrandomly;Weconsume;Weearninterestonourcash.Thefractionwonlychangesattheendofthetimestep,notduring(it’sratherliketheintheBlack–Scholesanalysis).Let’sbuildupthestochasticdifferential

7、equationforWinstages.Theeasiestbitistheconsumption.Wealthchangesbytheamountweconsume:dW=−C(t)dt+···.The···representallthetermstocome.Wegetinterestonthecash.Thecashweholdis(1−w)Wandsotheinterestreceivedisr(1−w)Wdt:dW=−C(t)dt+r(1−w)Wdt+···.Thefinalpartofthesto

8、chasticdifferentialequationisduetothechangeintheriskyasset.Sincetheassetpriceevolvesaccordingto(66.1)andweholdaproportionwofourwealthinthisasset,thereisachangeinourwealthofw(µWdt+σWdX).Notethattheasset